Quasi-isometric embeddings of symmetric spaces

This paper opens the study of quasi-isometric embeddings of symmetric spaces. The main focus is on the case of equal and higher rank. In this context some expected rigidity survives, but some surprising examples also exist. In particular there exist quasi-isometric embeddings between spaces $X$ and $Y$ where there is no isometric embedding of $X$ into $Y$ . A key ingredient in our proofs of rigidity results is a direct generalization of the Mostow–Morse lemma in higher rank. Typically this lemma is replaced by the quasiflat theorem, which says that the maximal quasiflat is within bounded distance of a finite union of flats. We improve this by showing that the quasiflat is in fact flat off of a subset of codimension $2$ .

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Keywords

symmetric spaces, quasi-isometries, coarse geometry, rigidity

Mathematical Subject Classification 2010

Primary: 22E40, 53C24, 53C35

References

Publication

Received: 24 May 2017

Revised: 14 January 2018

Accepted: 4 March 2018

Published: 1 June 2018

Proposed: Bruce Kleiner

Seconded: Martin Bridson, Anna Wienhard

Authors

David Fisher
Department of Mathematics Indiana University Bloomington, IN United States

Kevin Whyte
Department of Mathematics University of Illinois at Chicago Chicago, IL United States

Volume 22, issue 5 (2018)

David Fisher and Kevin Whyte

Abstract